As a continuation of a joint project for the development of exact statistical techniques for analyzing crystallographic problems, PSL has derived the characteristic function for the probabilistic analysis relevant to 206 out of the 230 possible crystallographic space groups. A second problem under current study is that of finding the probability density for the three phase invariant for space group P22. This involves a very heavy computational effort requiring analysis of the many symmetries inherent in the sixfold Fourier series representation. One further problem is that of finding a good approximation to the probability density of the three phase invariant for the P1 space group. Exact numerical results have been found at the expense of considerable computation. A much simpler approximation has been found that yields excellent agreement with numerical results.